fracture porosity of naturally fracture reservoir

Copyright 2006, Society of Petroleum Engineers This paper was prepared for presentation at the First International Oil Conference and Exhibition in Mexico held in Cancun, Mexico, 31 August–2 September 2006. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435. Abstract The storage capacity ratio, ω, measures the flow capacitance of the secondary porosity and the interporosity flow parameter, λ, is related to the heterogeneity scale of the system. Currently, both parameters λ and ω are obtained from well test data by using the conventional semilog analysis, type-curve matching or the TDS Technique. Warren and Root showed how the parameter ω can be obtained from semilog plots. However, no accurate equation is proposed in the literature for calculating fracture porosity. This paper presents an equation for the estimation of the λ parameter using semilog plots. A new equation for calculating the storage capacity ratio and fracture porosity from the pressure derivative is presented. The equations are applicable to both pressure buildup and pressure drawdown tests. The interpretation of these pressure tests follows closely the classification of naturally fractured reservoirs into four types, as suggested by Nelson 1 . The paper also discusses new procedures for interpreting pressure transient tests for three common cases: (a) the pressure test is too short to observe the early-time radial flow straight line and only the first straight line is observed, (b) the pressure test is long enough to observe the late-time radial flow straight line, but the first straight line is not observed due to inner boundary effects, such as wellbore storage and formation damage, and (c) Neither straight line is observed for the same reasons, but the trough on the pressure derivative is well defined. Analytical equations are derived in all three cases for calculating permeability, skin, storage capacity ratio and interporosity flow coefficient, without using type curve matching. In naturally fractured reservoirs, the matrix pore volume, therefore the matrix porosity is reduced as a result of large reservoir pressure drop due to oil production. This large pressure drop causes the fracture pore volume, therefore fracture porosity, to increase. This behavior is observed particularly in reservoir where matrix porosity is much greater than fracture porosity. Fractures in reservoirs are more vertically than horizontally oriented, and the stress axis on the formation is also essentially vertical. Under these conditions, when the reservoir pressure drops, the fractures do not suffer from the stress caused by the drop. Using these principles, a new method is introduced for calculating fracture porosity from the storage capacity ratio, without assuming the total matrix compressibility is equal to the total fracture compressibility. Several numerical examples are presented for illustration purposes. Introduction Nelson 1 identifies four types of naturally fractured reservoirs; based on the extent the fractures have altered the reservoir matrix porosity and permeability: In Type 1 reservoirs, fractures provide the essential reservoir storage capacity and permeability. Typical Type-1 naturally fractured reservoirs are the Amal field in Libya, Edison field California, and pre- Cambrian basement reservoirs in Eastern China. All these fields contain high fracture density. In Type 2 naturally fractured reservoirs, fractures provide the essential permeability, and the matrix provides the essential porosity, such as in the Monterey fields of California, the Spraberry reservoirs of West Texas, and Agha Jari and Haft Kel oil fields of Iran. In Type 3 naturally fractured reservoirs, the matrix has an already good primary permeability. The fractures add to the reservoir permeability and can result in considerable high flow rates, such as in Kirkuk field of Iraq, Gachsaran field of Iran, and Dukhan field of Qatar. Nelson includes Hassi Messaoud (HMD) in this list. While indeed there are several low- permeability zones in HMD that are fissured; in most zones however the evidence of fissures is not clear or unproven. In Type 4 naturally fractured reservoirs, the fractures are filled with minerals and provide no additional porosity or permeability. These types of fractures create significant reservoir anisotropy, and tend to form barriers to fluid flow and partition formations into relatively small blocks. Nelson discusses three main factors that can create reservoir anisotropy with respect to fluid flow: fractures, crossbedding and stylolite. The anisotropy in Hassi Messaoud field, for instance, appears to be the result of a non-uniform combination of all three factors with varying magnitude from zone to zone. Stylolites, just like fractures, are a secondary feature. They are defined as irregular planes of discontinuity between two rock units. Stylolites, which often have fractures associated with them, occur most frequently in limestone, SPE 104056 Fracture Porosity of Naturally Fractured Reservoirs D. Tiab, D.P. Restrepo, and A. Igbokoyi, SPE, U. of Oklahoma 2 SPE 104056 dolomite, and sandstone formations. Mineral-filled fractures and stylolites can create strong permeability anisotropy within a reservoir. The magnitude of such permeability is extremely dependent on the measurement direction, thereby requiring multiple-well testing. Interference testing is ideal for quantifying reservoir anisotropy and heterogeneity, because they are more sensitive to directional variations of reservoir properties, such as permeability, which is the case of type 4 naturally fractured reservoirs. It is important to take this classification into consideration when interpreting a pressure transient analysis for the purpose of identifying the type of fractured reservoir and its characteristics. Each type of naturally fractured reservoir may require a different development strategy. Ershaghi 2 reports that: (a) Type 1 fractured reservoirs, for instance, may exhibit sharp production decline and can develop early water and gas coning; (b) Recognizing that the reservoir is a type 2 will impact any infill drilling or the selection of improved recovery process; (c) In Type 3 reservoirs, unusual behavior during pressure maintenance by water or gas injection can be observed because of unique permeability trends. PROPERTIES OF MATRIX BLOCKS AND FRACTURES A naturally fractured reservoir is composed of a heterogeneous system of vugs, fractures, and matrix which are randomly distributed. Such type of system is modeled by assuming that the reservoir is formed by discrete matrix block elements separated by an orthogonal system of continuous and uniform fractures which are oriented parallel to the principal axes of permeability. Two key parameters, ω and λ, were introduced by Warren and Root 3 to characterize naturally fractured reservoirs. These dimensionless parameters λ and ω are mathematically expressed as 3 : mtft ft tt ft cc c c c )()( )( )( )( φφ φ φ φ ω + == ……………… (1) 2 2 m w f m x r k k αλ = …………………………………… (2) The geometry parameter, α, is defined as: )2(4 += nn α ………………………………… (3) where n is 1, 2 or 3 for the slab, matchstick and cube models, respectively. Assuming:(a) the flow between the matrix and the fractures is governed by the pseudo-steady state condition, but only the fractures feed the well at a constant rate, and (b) the fluid is single phase and slightly compressible, the wellbore pressure solution and the pressure derivative in an infinite- acting reservoir are given by 4,5 : s - t Ei - t Eit= P DD D D + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −++ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ )1()1( 80908.0ln 2 1 ω λ ωω λ … (4) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −× )1( exp )1( exp1 2 1 ' ωω λ ω λ - t + - t -= Pt DD DD ………… (5) The second pressure derivative of the dimensionless pressure equation is: ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − × )1( exp )1( exp 1 )1(2 )''( ω λ ωω λ ω ω λ - t - t = Pt DD DD …… (6) (A) Semilog Analysis A plot of the well pressure or pressure change (P) versus test time on a semilog graph should yield two parallel straight line portions as shown in Figure 1. The pressure change P during a drawdown test is (P i - P wf ). During a buildup test P = (P ws – P wf (t=0)). 1. Fracture Permeability Figure 1 shows two well defined parallel straight lines of slope m. The slope m of the straight lines may be used to calculate the average permeability of the fractured system or the k f h product: m qB kh o μ 6.162 = …………………………………. (7) Assuming the sugar cube model is valid and Types 1 naturally fractured reservoirs, the product kh is essentially equal to (kh) f , so the slope of either straight line can be used to determine kh. In Type 2 naturally fractured reservoirs the first straight line is mostly related to fracture flow, and therefore the kh product in Eq 7 is essentially (kh) f . The second straight line is however related to both fracture flow and matrix flow, thus the kh product in Eq 7 reflects both (kh) m and (kh) f . In this case it is unlikely that the two straight lines will be perfectly parallel. If however (kh) m << (kh) f then kh can be approximated by (kh) f . In Type 3 reservoirs, both straight lines are related to fracture flow and matrix flow, the product kh in Eq 7 is therefore equivalent to (kh) t . 2. Skin Factor The skin factor is obtained using conventional technique, i.e.: () ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − Δ = + 23.3log )( 1513.1 2 1 w mf t hr rc k m P s μφ (8) (P) 1hr is taken from the second straight line. 3. Fracture Storage Capacity Ratio The vertical distance between the two semilog straight lines, δP, may be used to estimate 3 the storage capacity ratio, ω: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= m P δ ω 303.2exp ………………………… (9) or mP / 10 δ ω − = ……………………………………… (10) In Type 4 naturally fractured reservoirs the value of  is close to unity. The sugar cube model is not realistic in Type 4 SPE 104056 3 fractured reservoirs, since the fractures do not provide additional porosity or permeability. These reservoirs are best treated as anisotropic and analyzed accordingly. 4. Interporosity Flow coefficient A characteristic minimum point, or trough, is typically observed on the pressure derivative plot for naturally fractured reservoirs, as shown in Figure 2. This minimum takes place at the point where the second pressure derivative equals zero (t D ×P D ’)’ = 0. The dimensionless time at which this minimum point occurs is given by the following expression 4, 5, 6 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ωλ ω 1 ln minD t …………………………… (11) On the semilog plot of well pressure versus test time, this minimum point corresponds to the inflection point during the transition portion of the curve. Therefore, Eq. 11 can be rewritten as: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ωλ ω 1 ln infD t ………………………………. (12) The dimensionless time is defined as: 2 inf inf )( 0002637.0 wmft D rc tk t μφ + = …………………………. (13) Where t inf = t min . Combining Eqs. 12 and 13 and solving for λ, yields a new relationship for the interporosity flow parameter: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + ω ω μφ λ 1 ln )(3792 inf 2 tk rc wmft ……………… (14) t inf can be directly read at the inflection point of the pressure curve from a semilog plot of the flowing well pressure versus test time. For a Miller-Dyes-Hutchinson (MDH) semilog plot, i.e. shut-in well pressure (P ws ) versus shut-in time (t), t inf = t inf . When using a Horner plot, the corresponding inflection (Horner) time, (HT) inf , is read and converted to inflection time using the following equation: 1)( inf inf − = T p H t t ……………………………… (15) Where (HT) is the Horner time (tp+t)/t or the effective Horner time tpt/(tp+t). The idea of estimating the interporosity flow parameter from semilog plots is not new. Uldrich and Ershaghi 7 , formulated a complex and cumbersome procedure for that purpose. They introduced one equation for pressure drawdown tests which uses the coordinates of the inflection point time, the storage capacity ratio, the skin factor and a parameter read from a plot which is a function of ω. They also introduced another equation for pressure buildup tests which utilizes the inflection point time, the storage capacity ratio, the dimensionless Horner production time, t D , and two parameters read from two different plots. These two graphically-obtained parameters are also function of the ω value. These equations have received limited applications. Bourdet and Gringarten 8 suggest plotting a horizontal line through the approximate middle of the transition portion of the curve, and then use the time at which this horizontal line intersects the parallel straight lines to calculate the storativity ratio, , and the interporosity flow coefficient, . Eq. 14 offers a much simpler and analytically sound procedure for calculating  from the conventional semilog analysis. 5. Short buildup Test – Second Straight is not observed The interpretation of a buildup test is similar to that of a drawdown. Generally, the second straight line is more likely to be observed than the first one, which often is masked by near wellbore effects, such as wellbore storage. In Type 3 naturally fractured system, where the matrix has a high enough permeability for the fluid to enter the wellbore both from the fracture (mostly) and the matrix, then the first straight line should last a long time, and will not be masked by inner wellbore effects. In this system, it is also possible for an unsteady state flow regime to develop in the matrix. This flow regime will appear during the transition period, i.e. after the first semilog straight line. However pressure buildup tests often give more reliable value of the storage capacity ratio, , especially when the second parallel straight line is not observed, such as when the pressure test is too short, or the well is near a boundary. In these cases it impossible to determine p, and consequently Eq. 10 can not be used. The equation of the early time straight line can be represented by 9 : ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ Δ+ −= ω ω 1 loglog t tt mPP p iws …… …… (16) Extrapolating the first straight line to a Horner time of unity, i.e. (t p +t)/t = 1, where P ws =P FF1 , then the storage capacity ratio can be calculated from: mPP mPP FFi FFi /)( /)( 1 1 101 10 − − − = ω … …………………….…….(17) P FF1 stands for “Fracture Flow” pressure, since near the wellbore, fluid flows into the well exclusively through the fractures, particularly in Types 1 and 2 naturally fractured reservoirs. P FF1 will always be greater than (by a value equal to p) the average pressure, P i and P*, since normally the second parallel line is used to estimate these three pressure values. If the initial reservoir pressure P i is not available, use the average reservoir pressure instead, or the false pressure P* (if it is known from another source). The vertical distance between the two parallel semilog straight lines and passing through the inflection point is of course identified as p. For uniformly distributed matrix 4 SPE 104056 blocks, the inflection point is at equal distance between the two parallel lines. Therefore m P inf1 2 10 Δ − = ω …………… ………………………… (18) Where: P 1inf (= 0.5P) is the pressure drop between the 1 st semilog straight line and the inflection point along a vertical line parallel to the pressure axis. Equation 18 is analogous to Eq. 10 for calculating the storage capacity ratio, and therefore should yield the same results as long as the first straight line is well defined and the pressure test is run long enough to observe the trough on the pressure derivative, and therefore the inflection point on the semilog plot. The interporosity flow coefficient is then calculated from Eq. 14. If the inflection point is difficult to determine, then read the end-time of the first or early time straight line, t EL1 , and use the following equation to estimate : ωω μφ λ )1( 013185.0 )( 1 2 − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + EL wmft kt rc ……… ……… (19) If the buildup test is however too short to even observe the trough (which provides the best evidence of a naturally fractured system), then results obtained from the interpretation of the test should at best be considered as an approximation. The skin factor is then obtained from the following equation: () ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − −−Δ = + 23.3log )()( 1513.1 2 11 w mf t FFihr rc k m PPP s μφ … (20) or () ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − Δ−Δ = + 23.3log 2)( 1513.1 2 inf11 w mf t hr rc k m PP s μφ ……. (21) where (P) 1hr is taken from the first straight line. EXAMPLE 1 Given the build up test data in Table 1 and the following formation and fluid properties, estimate formation permeability, skin factor, λ, and ω from. q = 125 STB/D h = 17 ft t p = 1200 hr φ = 13.0% p wf = 211.20 psia r w = 0.30 ft µ = 1.72 cp B=1.054 RB/STB c t =7.19×10 -6 psi -1 Solution The following data are read from Figure 3: t inf = 0.63 hr ΔP 1inf = 33 psi P 1hr = 497 psi m=35.67 psi/cycle t EL1 = 0.012 hr From Equation 7: mdk 7.60 )17)(67.35( )72.1)(054.1)(125(6.162 == From Equation 21 the storage capacity ratio is: 014.010 67.35 )33(2 == − ω Using equation 1, we can calculate (φc t ) f : 86 103.1 014.01 014.0 )1019.7)(13.0()( 1 )()( −− ×= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ×= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ft mtft c cc φ ω ω φφ From equation 21 the skin factor is: 89.0 23.3 )3.0)(72.1)(103.11019.713.0( 7.60 log 67.35 )3328.285( 1513.1 286 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ×+×× − ×− = −− s s From Equation 14, the interporosity flow parameter is: 7 286 107.8 014.0 1 ln014.0 )63.0)(7.60( )3.0)(72.1)(103.11019.713.0(3792 − −− ×= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ×× ×+×× = λ λ From Equation 19: 7 286 101.2 014.0)014.01( )012.0)(7.60)(013185.0( )3.0)(72.1)(103.11019.713.0( − −− ×= −× ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ×+×× = λ λ 6. Long buildup Test – First Straight is not Observed Generally, the second straight line is more likely to be observed than the first one, which often is masked by near wellbore effects, such as wellbore storage. In Type 1 and Type 2 naturally fractured systems, where the matrix permeability is negligible, the fluid flows into the wellbore exclusively through the fractures. The first straight line will probably be too short and easily masked by inner wellbore effects. The permeability and skin factor are calculated from Eqs. 7 and 8 respectively. The following equation provides a direct and accurate method for calculating , as long as the inflection point and the second straight line are observed and the matrix blocks are uniformly distributed: m P inf2 2 10 Δ − = ω ………… …………………………… (22) P 2inf (= 0.5p) is the pressure drop between the 2 nd semilog straight line and the inflection point along a vertical line parallel to the pressure axis. The interporosity flow parameters  is then calculated from Eq. 14. If the inflection point is difficult to determine, then read the starting-time of the second semilog straight line, t SL2 , and use the following equation to estimate : SPE 104056 5 )1( 1027.5 )( 2 5 2 ω μφ λ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × = − + SL wmft kt rc …… ………… …… (23) EXAMPLE 2 Given the build up test data in Table 2 and the following formation and fluid properties, estimate formation permeability, skin factor, λ, and ω. q = 125 STB/D h = 17 ft t p = 1200 hr φ = 13.0% p wf = 211.20 psia r w = 0.30 ft µ = 1.72 cp B=1.054 RB/STB. c t =7.19×10 -6 psi -1 Solution The following data are read from Figure 4: t inf = 3.05 hr ΔP 2inf = 24 psi P 1hr = 419 psi m=30 psi/cycle t SL2 =55 hr From Equation 7: mdk 25.72 )30)(17( )72.1)(054.1)(125(6.162 == From Equation 22: 025.010 30 )24(2 == − ω It is possible to calculate (φc t ) f by: 86 104.2 025.01 025.0 )1019.7)(13.0()( 1 )()( −− ×= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ×= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ft mtft c cc φ ω ω φφ From equation 8: ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ×+ − ×× −= 23.3 2 )3.0)(72.1)( 8 104.2 6 1019.713.0( 25.72 log 30 8.207 1513.1s 69.1=s From Equation 14: 7 286 1036.2 025.0 1 ln025.0 )05.3)(25.72( )3.0)(72.1)(104.21019.713.0(3792 − −− ×= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × ×+×× = λ λ From Equation 23: 7 5 286 1091.6 )025.01( )55)(25.72)(1027.5( )3.0)(72.1)(104.21019.713.0( − − −− ×= −× ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × ×+×× = λ λ (B) TDS Technique In 1993 Tiab introduced a technique 10 for interpreting loglog plots of the pressure and pressure derivative curves without using type curve matching. This technique utilizes the characteristic intersection points, slopes, and beginning and ending times of various straight lines corresponding to flow regimes strictly from loglog plots of pressure and pressure derivative data. Values of these points and slopes are then inserted directly in exact, analytical solutions to obtain reservoir and well parameters. This procedure for interpreting pressure tests, which is referred to as the Tiab’s Direct Synthesis (TDS) technique offers several advantages over the conventional semilog analysis and type curve matching. It has been applied to over fifty different reservoir systems 11-18 , and hundreds of field cases. 1. Fracture Permeability The pressure derivative portion corresponding to the infinite acting radial flow line is a horizontal straight line. This flow regime is given by 10 : kh Bq Pt R μ 6.70 )'( =Δ× …………………………… … (24) The subscript “R” stands for radial flow. The formation permeability is therefore: R Pth Bq k )'( 6.70 Δ× = μ …………………………… …… … (25) where (t×ΔP') R is obtained by extrapolating the horizontal line to the vertical axis. In order for the conventional semilog analysis and the TDS technique to yield the same value of k, the following equation must be true: R Ptm )'(303.2 Δ × = ……………….……………… (26) 2. Skin Factor The second radial flow line can also be used to calculate the skin factor from 10 : () ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − Δ× Δ = + 43.7 )( ln ' )( 5.0 2 2 2 2 wmft R R R rc kt Pt P s μφ ……… (27) Where t R2 is any convenient time during the system’s radial flow regime (as indicated by the horizontal line on the pressure derivative curve, Figure 2) and (ΔP) R2 is the value of ΔP on the pressure curve corresponding to t R2 . If the test is too short or the boundary is too close to the well to observe a well defined second straight line, then the skin factor can be estimated from the early-time horizontal straight line: 6 SPE 104056 () ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − Δ× Δ = + 43.7 1 )( ln ' )( 5.0 2 1 1 1 ω μφ wmft R R R rc kt Pt P s …… (28) Where t R1 is any convenient time during the early-time radial flow regime (as indicated by the horizontal line on the pressure derivative curve, Figure 2) and (ΔP) R1 is the value of ΔP on the pressure curve corresponding to t R1 . 3. Interporosity Flow Coefficient The interporosity flow parameter can also be obtained from the loglog plot of the derivative function (txP’) versus test time 4,5 by substituting the coordinates of the minimum point of the trough, t min and (txP’) min : () min min 2 ' )(5.42 t Pt qB rch o wmft Δ× ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = + φ λ …………….… (29) The advantage of Eq. 29 over Eq. 14 is that it is independent of permeability and storage capacity ratio, and the coordinates of the minimum points are easier to determine than the inflection point on the semilog plot. 4. Storage Capacity Ratio The coordinates of the minimum point of the trough can be used to derive two equations to calculate accurately the storage capacity ratio . Pressure derivative Coordinate: Using the pressure derivative coordinate of the minimum point and the radial flow regime (horizontal) line, the following equation provides a direct and accurate method for calculating : ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ× Δ× −− = R Pt Pt )'( )'( 18684.0 min 10 ω …… …………….…… (30) Equation 30 is derived by observing that: 2 )()( min P PtPt R δ = ′ Δ×− ′ Δ× ………………… … (30a) Combining Equations 30a and 26 yields: () ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ Δ× ′ Δ× −= ′ Δ× ′ Δ×− ′ Δ× = R R R Pt Pt Pt PtPt m P )( )( 18684.0 )(303.2 )()(2 min min δ ………….……… (30b) Substituting Equation 30b into Equation 10 yields Equation 30. Equation 30 assumes wellbore storage and boundary effects do not influence the trough and the infinite acting radial flow line is well defined. In conventional analysis this ideal case displays two well defined parallel lines with the inflection point equidistant of those two lines, which means that the fractures are uniformly distributed. Minimum Time: Using the time coordinate of the minimum point, a less direct but just as accurate value of the storage capacity ratio can be obtained when wellbore storage is present from the following equation: minD t e λ ω ω − = …………………………… ………. (31) Where the dimensionless time at the minimum point is calculated from: min 2 min )( 0002637.0 t rc k t wmft D ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + μφ …………… …………. (32) Solving explicitly for  Eq. 31 yields 19 : 1 5452.6 )ln( 5688.3 9114.2 − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−= SS NN ω …………………. (33) Where the parameter N S is given by: minD t S eN λ − = ……………………………………… (34) Eq. 34 is obtained by assuming values of , from 0 to 0.5, then values of   = N S were plotted against . The resulting curve was curve-fitted. Note that Eq. 33 can also be used in the semilog analysis since t min = t inf . It is recommended that both methods be used for comparison purposes. If the radial flow regime line on the derivative curve is not well defined due to a combination of inner and/or outer boundary effects or a short test, but the minimum of the trough is well defined, then Eqs. 29 and 33 should be used to calculate, respectively,  and . EXAMPLE 3 Tiab Direct Synthesis technique is applied to Example 2. Figure 5 is plotted with data from Table 2 and the respective pressure derivative. From Figure 5 the following data can be read: P R = 274.51 psi t R = 156.51 psi t min = 3.05 hours (t×P’) min = 1.3 psi (t×P’) R = 13 psi t e =0.018 hr P e = 13 psi Wellbore storage coefficient is calculated by 10 : psibbl P t qB C e /1060.7 13 018.0 24 )054.1)(125( 24 3− ×= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ = From Equation 25: mdk 39.72 )13)(17( )054.1)(72.1)(125(6.70 == From Equation 27: 74.1 43.7 )3.0)(72.1)(104.21019.713.0( )51.156)(39.72( ln 13 51.274 5.0 286 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ×+×× −= −− s s SPE 104056 7 From Equation 29: 7 286 1002.2 05.3 3.1 )054.1)(125( )3.0)(104.21019.713.0)(17(5.42 − −− ×= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ×+×× = λ λ From Equation 33 the storage capacity ratio can be calculated in the presence of wellbore storage: 024.0 924.0 5452.6 )924.0ln( 5688.3 9114.2 1 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−= − ω Table 3 is a comparison of the TDS results with that of conventional method. FRACTURE POROSITY AND COMPRESSIBILTY Once ω is estimated, the fracture porosity can be estimated if matrix porosity, φ m , total matrix compressibility, c tm , and total fracture compressibility, c tf , are known, as follows: m tf tm f c c φ ω ω φ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 1 ………………………… ……… (35) Fracture compressibility may be different from matrix compressibility by an order of magnitude. Naturally fractured reservoirs in Kirkuk field (Iraq) and Asmari field (Iran) have fracture compressibility ranging from 4x10 -4 to 4x10 -5 psi -1 . In Grozni field (Russia) c tf ranges from 7x10 -4 to 7x10 -5 . In all these reservoirs c tf is 10 to 100 folds higher than c tm . Therefore the practice of assuming c tf = c tm is not acceptable. The fracture compressibility can be estimated from the following expression 9 : () () P kk P kk c fiffif tf Δ − ≈ Δ − = 3 /1/1 3/1 … ……………… (36) = fi k Fracture permeability at the initial reservoir pressure i p = f k Fracture permeability at the current average reservoir pressure. Combining Equations 35 and 36 yields 19 : () 3/1 )/(1 1 fif tmmf kk P c − Δ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ω ω φφ ………………… (37) In deep naturally fractured reservoirs, fractures and the stress axis on the formation generally are vertically oriented. Thus when the pressure drops due to reservoir depletion, the fracture permeability reduces at a lower rate than one would expect. In Type-2 naturally fractured reservoirs, where matrix porosity is much greater than fracture porosity, as the reservoir pressure drops the matrix porosity decreases in favor of fracture porosity 9 . This not the case in Type-1 naturally fractured reservoirs, particularly if the matrix porosity is very low or negligible. For fractured reservoirs and, indeed, all highly anisotropic reservoirs, the geometric mean is currently considered the most appropriate of the three most common averaging techniques (arithmetic, harmonic and geometric). Therefore, a representative average value of the effective permeability of a naturally fractured reservoir may be obtained from the geometric mean of k max and k min as illustrated in Figure 6. minmax kkk = ……………… …………………… (38) where k max = maximum permeability measured in the direction parallel to the fracture plane (Figure 6), thus k max ≈ k fracture k min = minimum permeability measured in the direction perpendicular to the fracture plane (Figure 6), thus k min ≈ k matrix Substituting k f and k m for, respectively, k max and k min , Equation 38 becomes: mf kkk = ………………………………………… (39) The fracture permeability can therefore be estimated from: m f k k k 2 = …………………………………………… (40) Where k m is the matrix permeability, which is measured from representative cores and k is the mean permeability obtained from pressure transient tests. Combining equations 36 and 40 yields: ( ) P kk c i tf Δ − = 3/2 /1 ………………………………… (41) Where k i = average permeability obtained from a transient test run when the reservoir pressure was at or near initial conditions P i and k = average permeability obtained from a transient test at the current average reservoir pressure. PPP i −=Δ Combining Equations 41 and 35 yields 19 : () 3/2 )/(1 1 i tmmf kk P c − Δ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ω ω φφ ……… … …… (42) Matrix permeability is assumed to remain constant between the two tests. Note that equations 37 and 42 are also valid for calculating fracture porosity change between two consecutive pressure transient tests, and therefore 21 PPP −=Δ . The time between the two tests must be long enough for the fractures to deform significantly in order to determine an accurate value of c tf . Table 5 shows pressure transient analysis in Cupiaga field, a naturally fractured reservoir in Colombia 22 . The reduction in permeability for well 1 is about 13% and the change in pressure is 344 psi from 1996 to 1997. This type of data can be used in order to estimate φ f from Eq. 42. Eq. 37 should yield a more accurate value of fracture porosity than Eq. 42, as the latter assumes Eq. 39 is always applicable. Substituting the values of ffm andkk φ ,, into the following equation should yield approximately the same value of the effective permeability obtained from well testing 20 : ffm kkk φ + ≈ …………………… ………………… (43) 8 SPE 104056 Eq. 43 should only be used for verification purposes. The fracture width or aperture may be estimated 20 from t f f k w ωφ 33 = ……………… …………………….…. (44) where: fracture width = microns, permeability = mD, porosity = fraction, and storage capacity = fraction. EXAMPLE 4 Pressure tests in the first few wells located in a naturally fractured reservoir yielded a similar average permeability of the system of 82.5 mD. An interference test also yielded the same average reservoir permeability, which implies that fractures are uniformly distributed. The total storativity, (φc t ) m+f = 1x10 -5 psi -1 was obtained from this interference test. Only the porosity, permeability and compressibility of the matrix could be determined from the recovered cores. The pressure data for the well are given in Table 4. The pressure drop from the initial reservoir pressure to the current average reservoir pressure is 300 psi. The characteristics of the rock, fluid and well are given below: q = 3000 STB/D h = 25 ft φ m = 10% r w = 0.4 ft µ = 1 cp B=1.25 RB/STB. c tm =1.35×10-5 psi -1 k m =0.10 mD 1 - Using conventional semilog analysis and TDS technique, calculate the current formation permeability, storage capacity ratio, and fluid transfer coefficient 2 – Estimate the three fracture properties: permeability, porosity and width. Solution 1(a) – Conventional method From Figure 7: δP = 130 psi m=325 psi/cycle t inf =2.5 hrs The average permeability of the formation is estimated from the slope of the semilog straight line. Using Equation 7 yields: ()()() ()() mDk 05.75 25325 125.130006.162 == Fluid storage coefficient is estimated using Equation 10: 39.010 )325/130( == − ω The storage coefficient of 0.39 indicates that the fractures occupy 39% of the total reservoir pore volume. The inter-porosity fluid transfer coefficient is given by Equation 14: ( ) ()() 5 25 1019.1 39.0 1 ln39.0 5.205.75 )4.0)(1(1013792 − − ×= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × = λ 1(b) – TDS technique From Figure 8, the following characteristic points are read: Δt min = 2.5 hrs (t×ΔP’) R = 146 psi (t×ΔP’) min = 70.5 psi Using the TDS technique, the value of k is obtained from Equation 25: ( ) ( ) ( ) ()( ) mDk 53.72 14625 25.1130006.70 == The inter-porosity fluid transfer coefficient is given by Equation 29: 5 25 1028.1 5.2 )5.70( )25.1)(3000( )4.0)(101)(25)(5.42( − − ×= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × = λ Since the two parallel lines are well defined the storage coefficient ω is calculated from Equation 30 35.010 146 5.70 18684.0 == ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −− ω The conventional semilog analysis yields similar values of k,  and  as the TDS technique. The main reason for this match is that both parallel straight lines are well defined. 2 – Current properties of the fracture (a) The fracture permeability is calculated from Equation. 40: mD k k k m f 606,52 10.0 53.72 22 === The fracture permeability at initial reservoir pressure is: mD k k k m i fi 062,68 10.0 5.82 2 2 === (b) The fracture porosity In fractured reservoirs with deformable fractures, the fracture compressibility changes with declining pressure. The fracture compressibility can be estimated from Equation 41: ( ) 14 3/2 102.5 300 062,68/606,521 −− ×= − = psic tf The compressibility ratio is: 5.38 1035.1 102.5 5 4 = × × = − − tm tf c c Thus, the fracture compressibility is more than 38.5 folds higher than the matrix compressibility, or tmtf cc 5.38 = . The fracture porosity from Equation 42 is: %14.000139.0 5.38 1.0 35.01 35.0 ≈= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = f φ The total porosity of this naturally fractured reservoir is: 1014.00014.010.0 = + = + = fmt φ φ φ Substituting the values of ffm andkk φ ,, into Equation 43: mDkkk ffm 7.73606,520014.01.0 =× + = + ≈ φ This value is approximately the same value of the effective permeability obtained from well testing (72.53 mD). The fracture width or aperture may be estimated from Equation 44: mmmicronsw f 212.0212 1014.035.033 606,52 == ×× = SPE 104056 9 The fracture width is a useful parameter for identifying the nature of fracturing in the reservoir. Conclusions 1. The inflection point on the semilog plot of well pressure versus test time and the corresponding minimum point on the trough of the pressure derivative curve are unique points that can be used to characterize a naturally fractured reservoir. 2. The interporosity flow parameter can be accurately obtained from the conventional semilog analysis if the inflection point is well defined and the new proposed equation is utilized. The equation is valid for both pressure drawdown and pressure buildup tests. 3. Two new equations are introduced for accurately calculating the storage capacity ratio from the coordinates of the minimum point of the trough on the pressure derivative curve. 4. For a short test, in which the late-time straight line is not observed, the storage capacity ratio and the interporosity flow coefficient can both be calculated from the inflection point. 5. For a long test, in which the early-time straight line is not observed, due to near-wellbore effects, the storage capacity ratio can also be calculated from the inflection point. 6. A new equation is proposed for calculating fracture porosity, as a function of reservoir compressibility. 7. The practice of assuming the total compressibility of the matrix (c tm ) is equal to the total compressibility of the fracture (c tf ) should be avoided. From field observations, c tf is several folds higher than c tm . Nomenclature B oil volumetric factor, rb/STB c system compressibility, psi -1 h formation thickness, ft H T Horner time, dimensionless k permeability, md m semilog slope, psi/log cycle P ws well shutin pressure, psi P wf well flowing pressure, psi q oil flow rate, BPD r w wellbore radius, ft s skin factor t p producing time before shut-in, hrs w f Fracture width in microns Greek Symbols δP vertical distance between the two semilog straight lines, psi α Geometry parameter, 1/L2 φ Porosity, dimensionless ΔP 1inf Pressure drop between the 1 st semilog strigth line and the inflection point, psi ΔP 2inf Pressure drop between the 2 nd semilog strigth line and the inflection point, psi Δt shut-in time, hrs λ Interporosity flow parameter, dimensionless µ Viscosity, cp ω Storage capacity ratio, dimensionless Subscripts i initial o oil D dimensionless f fracture, fissure m matrix t total inf inflection point min minimum 1 1st semilog straight line 2 2nd semilog straight line 1hr 1 hour References 1. Nelson, R.: “Geologic Analysis of Naturally Fractured Reservoirs”. Gulf Professional Publishing, 2 nd Edition. 2001 2. Ershaghi, I.: “Evaluation of Naturally Fractured Reservoirs”. IHRDC, PE 509, 1995. 3. Warren, J.E. and Root, P.J.: “The Behavior of Naturally Fractured Reservoirs”. Soc. Pet. Eng. J. (Sept. 1963): 245-255. Trans. AIME, 228. 4. Engler, T. and Tiab, D.: “Analysis of Pressure and Pressure Derivative without Type Curve Matching, 2. Naturally Fractured Reservoirs”. Journal of Petr. Sci. and Eng. 15 (1996):127-138. 5. Engler, T. and Tiab, D.: “Analysis of Pressure and Pressure Derivative without Type Curve Matching, 5. Horizontal Well Tests in Naturally Fractured Reservoirs”. Journal of Petr. Sci. and Eng. 15 (1996); 139-151. 6. Engler, T. and Tiab, D.: “Analysis of Pressure and Pressure Derivative without Type Curve Matching - 6. Horizontal Well Tests in Anisotropic Media”. Journal of Petroleum Science and Engineering, Vol. 15 (Aug. 1996) N 0 . 2-4, 153-168. 7. Uldrich, D.O. and Ershaghi, I.: “A Method for Estimating the Interporosity Flow Parameter in Naturally Fractured Reservoirs”: Paper SPE 7142, Proceedings, 48 th SPE-AIME Annual California Regional Meeting held in San Francisco, CA, Apr. 12-14, 1978. 8. Bourdet, D. and Gringarten. AC.: “Determination of fissured volume and block size in fractured reservoirs by type-curve analysis”. Paper SPE 9293. Soc. Pet. Eng., Annu. Tech. Conf., Dallas, TX, Sept. 21-24, 1980, 9. Saidi, M. A.: “Reservoir Engineering of Fractured Reservoirs”. Total Edition Presse, 1987. 10. Tiab, D.: "Analysis of Pressure and Pressure Derivative without Type-Curve Matching - 1. Skin and Wellbore Storage". Journal of Petroleum Science and Engr., Vol. 12, No. 3 (January, 1995) 171-181. 11. Jongkittinarukorn, K. and Tiab, D.: “Analysis of Pressure and Pressure Derivative without Type Curve Matching - 6. Vertical Well in Multi-boundary Systems”. Proceedings, CIM 96-52, 47th Annual Tech. Meeting, Calgary, Canada, June 10-12, 1996. 12. Jongkittinarukorn, K. and Tiab, D.: “Analysis of Pressure and Pressure Derivatives without Type Curve Matching - 7. Horizontal Well in a Closed Boundary Systems”, Proceedings, CIM 96-53, 47th Annual Tech. Meeting, Calgary, Canada, June 10-12, 1996. 13. Tiab, D., Azzougen, A., F.H., Escobar, and S. Berumen: “Analysis of Pressure Derivative Data of Finite-Conductivity Fractures by the Tiab’s Direct Synthesis Technique”. Paper SPE 52201. Proceedings, SPE Mid-Continent Operations Symposium, Oklahoma City, 28 – 31 March 1999; Proceedings 10 SPE 104056 SPE Latin American & Caribbean Petr. Engr. Conf., Caracas, Venezuela, 21–23 April 1999, 17 pages. 14. Mongi, A. and Tiab, D.: “Application of Tiab’s Direct Synthesis Technique to Multi-rate Tests”, SPE/AAPG 62607, Proceedings, Western Regional Meeting, Bartlesville, California, 19-23 June 2000. 15. Benaouda, A. and Tiab, D.: “Application of Tiab’s Direct Synthesis Technique to Gas Condensate Wells”. Proceedings, SPE Permian Basin Conference, Texas, May 2001 16. Jokhio, S.A., Hadjaz, A. and Tiab, D.: “Pressure falloff Analysis in Water Injection Wells Using the Tiab’s Direct Synthesis Technique”. Paper SPE 70035, Proceedings, SPE Permian Basin Conference, Midland, Texas, May 15-16, 2001. 17. Bensadok A. and Tiab, D.: “Interpretation of Pressure Behavior of a Well between Two Intersecting Leaky Faults Using Tiab’s Direct Synthesis (TDS) Technique”. CIP2004-123, Proceedings, Canadian International Petroleum Conference, 7 – 10 June 2004 18. Chacon, A., Djebrouni, A. and Tiab, D.: “Determining the Average Reservoir Pressure from Vertical and Horizontal Well Test Analysis Using Tiab’s Direct Synthesis Technique”. Paper SPE 88619, Proceedings, Asia Pacific Oil and Gas Conference and Exhibition, Perth, Australia, Oct. 18-20, 2004. 19. Tiab, D. and E.C. Donaldson: “Petrophysics: theory and practice of measuring reservoir rock and fluid transport properties”. Gulf professional Publications, 2 nd Edition, 2004. 20. Bona, N., Radaelli, F., Ortenzi, A., De Poli, A., Pedduzi, C. and Giorgioni, M: “Integrated Core Analysis for Fractured Reservoirs: Quantification of the Storage and Flow Capacity of Matrix, Vugs, and Fractures”. SPERE, Aug. 2003, Vol.6, pp.226-233. 21. Stewart G. Ascharsobbi F. “Well test interpretation for Naturally Fractured Reservoirs”. Paper SPE 18173. 22. Giraldo L. A., Chen Her-Yuan, Teufel L. W. “ Field Case Study of Geomachanical Impact of Pressure Depletion in the Low- Permeability Cupiaga Gas-Condensate Reservoir”. SPE 60297. SPE Rocky Mountain Regional/Low Permability Reservoirs Symposium, Denve, CO, March 12-15, 200. SI Metric Conversion Factors bbl x 1.589873 E-01 = m 3 cp x 1.0 * E-03 = Pa-s ft x 3.048 * E-01 = m ft 2 x 9.290304 E-02 = m 2 psi x 6.894757 E+00 = kPa *Conversion factor is excat. Table 1. Pressure data for Example 1 Time hours Pressure psi P psi Horner Time 0.0000 211.20 0.00 0.0010 390.73 179.53 1200001.00 0.0023 404.32 193.12 521740.13 0.0040 413.00 201.80 300001.00 0.0062 419.73 208.53 193549.39 0.0090 425.39 214.19 133334.33 0.0128 430.36 219.16 93751.00 0.0176 434.81 223.61 68182.82 0.0239 438.82 227.62 50210.21 0.0320 442.43 231.23 37501.00 0.0426 445.66 234.46 28170.01 0.0564 448.48 237.28 21277.60 0.0743 450.87 239.67 16151.74 0.0976 452.84 241.64 12296.08 0.1279 454.36 243.16 9383.33 0.1673 455.46 244.26 7173.74 0.2190 456.20 245.00 5480.45 0.2850 456.65 245.45 4211.53 0.3720 456.90 245.70 3226.81 0.4840 457.03 245.83 2480.34 0.6300 457.11 245.91 1905.76 0.8200 457.18 245.98 1464.41 1.0670 457.27 246.07 1125.65 1.3890 457.39 246.19 864.93 1.8060 457.55 246.35 665.45 2.3500 457.75 246.55 511.64 3.0500 458.01 246.81 394.44 [...]... Minimum point tR2 10 100 1000 10000 Time, hr Figure 2 – P and pressure derivative plot for a naturally fractured reservoir Inflection point m 6,100 6,000 5,900 5,800 t H-inf 10,000 P)R2 P)R1 10 6,200 5,700 100,000 1000 1,000 Horner time, 100 10 tH= (tp + t)/ t Figure 1- Semilog pressure behavior of a naturally fractured reservoir 12 SPE 104056 W 520 Inflection point t inf= 0.63 hr Shut in pressure, psi 500... impact on relative permeability is less close to the wellbore than deeper in the reservoir Homogeneous reservoir model P*is difficult to estimate because of the variation of Kh with radius Given that the pressure at 100 ft radius is well above dew point, it is of some concern that Kh has not been fully restored At a radius of 350 ft the Kh reduces below 2000 md-ft ... observed The effects of the condensate banking are observed in the GOR response, at higher drawdowns the GOR increased 1996 Well 1 5660 3410 14.2 20 171 Post-Frac Test Homogeneous reservoir model Some drainage area is still above the dew point 1997 Well 1 5150 3000 4.3 5 171 Drainage area Below dew-point Homogeneous reservoir model Derivative curve indicates radial flow with a low value of gas effective... Analysis of Selected Cupiaga wells22 Test Type P* Global Particular K (md) (psi) Date BHFP Skin Types Comments/Remarks h(ft) of Skin Well 1 6004 91.3 Mechanical 16.4 48 171 Mechanical 16.4 18 3000 Pre-Frac Test 171 Homogenous reservoir model The turbulence factor is quite large due to non-darcy flow (high rates) combined with the condensate banking 1996 Well 1 6004 3200 38 Post-Frac Test Homogeneous reservoir. .. of 800 ft followed by an increase in effective permeability further out This is interpreted as being due to liquid condensate drop-out 1998 Well 1 5050 5380 3 Injector PFO 6267 Three layer model 4600 2772 Total Total Mechanical 8.35 0 429 1 Includes Turbulent and Condensate effects 0.6 1998 Well 2 Homogeneous reservoir model A small negative mechanical skin is suggested possibly due to activation of. .. model A small negative mechanical skin is suggested possibly due to activation of fractures by injecting pressure/temperature The well is in an under-injecting situation 11.4 19 3100 1995 Well 2 171 than non -Darcy 1998 Well 2 Zero skin for other 5500? 6947 1(@100 ft) 993 2.9 Injector PFO 993 1999 (@100 ft) Homogeneous reservoir model Entire drainage area is below dew point pressure The explanation . causes the fracture pore volume, therefore fracture porosity, to increase. This behavior is observed particularly in reservoir where matrix porosity is much greater than fracture porosity. Fractures. naturally fractured reservoirs, where matrix porosity is much greater than fracture porosity, as the reservoir pressure drops the matrix porosity decreases in favor of fracture porosity 9 method. FRACTURE POROSITY AND COMPRESSIBILTY Once ω is estimated, the fracture porosity can be estimated if matrix porosity, φ m , total matrix compressibility, c tm , and total fracture